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\fancyhead[L]{\small{\textbf{Gruppe 6, Udo Schlegel, Sirui Liu, Blatt 4}}}
\begin{document}
\section{Aufgabe 1}
$M_{per} * P = M_{orth} \Rightarrow M_{per}^{-1} * M_{per} * P = M_{per}^{-1} * M_{orth} \Rightarrow P = M_{per}^{-1} * M_{orth}$\\
$[M_{per}|E]=$
$
\left[
\begin{matrix} \frac{2n}{r-l}&0&\frac{r+l}{r-l}&0\\ 0&\frac{2n}{t-b}&\frac{t+b}{t-b}&0\\ 0&0&\frac{-(f+n)}{f-n}&\frac{-2fn}{f-n}\\0&0&-1&0 \end{matrix} 
\left| 
\begin{matrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{matrix}
\right] \right.
$  
\\
$
=\left[
\begin{matrix} \frac{2n}{r-l}&0&\frac{r+l}{r-l}&0\\ 0&\frac{2n}{t-b}&\frac{t+b}{t-b}&0\\ 0&0&0&\frac{-2fn}{f-n}\\0&0&-1&0 \end{matrix} 
\left| 
\begin{matrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\frac{f-n}{-2fn}&\frac{f+n}{f-n}\\ 0&0&0&1 \end{matrix}
\right] \right.
$  
\\
$
=\left[
\begin{matrix} \frac{2n}{r-l}&0&\frac{r+l}{r-l}&0\\ 0&\frac{2n}{t-b}&\frac{t+b}{t-b}&0\\ 0&0&0&1\\0&0&-1&0 \end{matrix} 
\left| 
\begin{matrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\frac{f-n}{-2fn}&\frac{-(f+n)}{2fn}\\ 0&0&0&1 \end{matrix}
\right] \right.
$  
\\
$
=\left[
\begin{matrix} \frac{2n}{r-l}&0&\frac{r+l}{r-l}&0\\ 0&\frac{2n}{t-b}&0&0\\ 0&0&0&1\\0&0&-1&0 \end{matrix} 
\left| 
\begin{matrix}1&0&0&0\\ 0&1&0&\frac{t+b}{t-b}\\ 0&0&\frac{f-n}{-2fn}&\frac{-(f+n)}{2fn}\\ 0&0&0&1 \end{matrix}
\right] \right.
$  
\\
$
=\left[
\begin{matrix} \frac{2n}{r-l}&0&\frac{r+l}{r-l}&0\\ 0&1&0&0\\ 0&0&0&1\\0&0&-1&0 \end{matrix} 
\left| 
\begin{matrix}1&0&0&0\\ 0&\frac{t-b}{2n}&0&\frac{t+b}{2n}\\ 0&0&\frac{f-n}{-2fn}&\frac{-(f+n)}{2fn}\\ 0&0&0&1 \end{matrix}
\right] \right.
$  
\\
$
=\left[
\begin{matrix} \frac{2n}{r-l}&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\0&0&-1&0 \end{matrix} 
\left| 
\begin{matrix}\frac{r-l}{2n}&0&0&\frac{r+l}{r-l}\\ 0&\frac{t-b}{2n}&0&\frac{t+b}{2n}\\ 0&0&\frac{f-n}{-2fn}&\frac{-(f+n)}{2fn}\\ 0&0&0&1 \end{matrix}
\right] \right.
$  
\\
$
=\left[
\begin{matrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\0&0&0&1 \end{matrix} 
\left| 
\begin{matrix}\frac{r-l}{2n}&0&0&\frac{r+l}{2n}\\ 0&\frac{t-b}{2n}&0&\frac{t+b}{2n}\\  0&0&0&-1\\ 0&0&\frac{f-n}{-2fn}&\frac{-(f+n)}{2fn} \end{matrix}
\right] \right.
$  
\\
$= [E|M_{per}^{-1}]$\\
$M_{per}^{-1} = \begin{bmatrix}\frac{r-l}{2n}&0&0&\frac{r+l}{2n}\\ 0&\frac{t-b}{2n}&0&\frac{t+b}{2n}\\  0&0&0&-1\\ 0&0&\frac{f-n}{-2fn}&\frac{-(f+n)}{2fn} \end{bmatrix}$
\\
$\Rightarrow$
\\
$P = M_{per}^{-1} * M_{orth}$
\\
$\begin{bmatrix}\frac{r-l}{2n}&0&0&\frac{r+l}{2n}\\ 0&\frac{t-b}{2n}&0&\frac{t+b}{2n}\\  0&0&0&-1\\ 0&0&\frac{f-n}{-2fn}&\frac{-(f+n)}{2fn} \end{bmatrix} * 
\begin{bmatrix}
\frac{2}{r-l}&0&0&\frac{-(r+l)}{r-l}\\
0&\frac{2}{t-b}&0&\frac{-(t+b)}{t-b}\\
0&0&\frac{-2}{f-n}&\frac{-(n+f)}{f-n}\\
0&0&0&1
\end{bmatrix}$
\\
$= \begin{bmatrix}
\frac{1}{n}&0&0&0\\
0&\frac{1}{n}&0&0\\
0&0&0&-1\\
0&0&\frac{1}{fn}&0
\end{bmatrix}
$
\end{document}